On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups

Details On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups

2010-02-25

arxiv.org/abs/1002.4792v2

Mathematics

Representation Theory

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Scientific paper

Let $G$ be a 1-connected Banach-Lie group or, more generally, a BCH--Lie group. On the complex enveloping algebra $U_\C(\g)$ of its Lie algebra $\g$ we define the concept of an analytic functional and show that every positive analytic functional $\lambda$ is integrable in the sense that it is of the form $\lambda(D) = \la \dd\pi(D)v, v\ra$ for an analytic vector $v$ of a unitary representation of $G$. On the way to this result we derive criteria for the integrability of *-representations of infinite dimensional Lie algebras of unbounded operators to unitary group representations. For the matrix coefficient $\pi^{v,v}(g) = \la \pi(g)v,v\ra$ of a vector $v$ in a unitary representation of an analytic Fr\'echet-Lie group $G$ we show that $v$ is an analytic vector if and only if $\pi^{v,v}$ is analytic in an identity neighborhood. Combining this insight with the results on positive analytic functionals, we derive that every local positive definite analytic function on a 1-connected Fr\'echet--BCH--Lie group $G$ extends to a global analytic function.

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